Question

Find the area of the region bounded by the curves y = x2 and y = cos(x). Give your answer correct to 2 decimal places

Answer 1

1.09 square units.

Explanation:

The given curves are

`y=x^2\\\\y=\cos x`

Let us plot the graph of these curves in the same coordinate axes. The graphs and the shaded region has been shown in the attached image.

The area of shaded region is given by

`A=2\int_0^(0.824)\left(\cos x-x^2\right)dx`

Now, we evaluate this definite integral to find the area of the shaded region.

`2\left [ \sin x-(x^3)/(3) \right ]_(0)^(0.824)\\\\2*(0.73386-0.18649)\\\\=2\cdot \:0.54737\\\\=1.09475`

Therefore, the area of the region bounded by the given curves is 1.09 square units.

Answer 2

You want to integrate the function
.. f(x) = cos(x) -x^2
between its zeros, which are near ±0.8241. These zeros can be determined from a graph of the function.

Once the integration limit is found, the function can be integrated analytically to get
.. area = 2(sin(0.8241) -(1/3)(0.8241)^3) ≈ 1.09